To evaluate [tex]\(\cos \left(\sin ^{-1}\left(\frac{1}{2}\right)\right)\)[/tex], let's proceed step-by-step.
1. Evaluate the Inverse Sine:
[tex]\[
\theta = \sin^{-1}\left(\frac{1}{2}\right)
\][/tex]
By definition, [tex]\(\theta\)[/tex] is the angle whose sine is [tex]\(\frac{1}{2}\)[/tex].
2. Identify the Angle for the Sine Value:
The angle [tex]\(\theta\)[/tex] that satisfies [tex]\(\sin \theta = \frac{1}{2}\)[/tex] within the principal range of [tex]\(\sin^{-1}\)[/tex] (i.e., [tex]\(-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}\)[/tex]) is:
[tex]\[
\theta = \frac{\pi}{6} \quad \text{(or approximately } 0.5236 \text{ radians)}
\][/tex]
3. Evaluate the Cosine of the Angle:
Now we need to find:
[tex]\[
\cos \left(\frac{\pi}{6}\right)
\][/tex]
4. Known Cosine Value:
Using the known trigonometric values for the cosine function, we have:
[tex]\[
\cos \left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \approx 0.8660254037844386
\][/tex]
Therefore, the evaluation of [tex]\(\cos \left(\sin^{-1}\left(\frac{1}{2}\right)\right)\)[/tex] is:
[tex]\[
\cos \left(\sin^{-1}\left(\frac{1}{2}\right)\right) \approx 0.8660254037844386
\][/tex]
So, the final result is:
[tex]\[
0.8660254037844386
\][/tex]
